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Chapter 4: Problem 23

Find the next three terms of each sequence. \(13,14,16,19, \ldots\)

Short Answer

Expert verified
The next three terms are 23, 28, and 34.

Step by step solution

01

Identify the Patterns Between Terms

Observe the differences between successive terms in the sequence. The given sequence is 13, 14, 16, 19. - Difference between 14 and 13 is 1. - Difference between 16 and 14 is 2. - Difference between 19 and 16 is 3. These differences suggest that the differences themselves increase by 1 each time.
02

Continue the Pattern of Differences

Since the difference between successive terms increases by 1 each time, the next difference should be 4. Add this to the last term given in the sequence, which is 19: 19 + 4 = 23.
03

Calculate the Next Term in the Sequence

Based on the pattern established, increase the difference by 1 once more to get the next number in the sequence. The new difference will be 5: 23 + 5 = 28.
04

Calculate the Final Term to Find

Following the sequence, the difference increases by 1 again to become 6. Add this to the last calculated term: 28 + 6 = 34.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pattern Recognition
Understanding patterns is key to solving arithmetic sequences like the one we are dealing with here. In the given sequence, the numbers are 13, 14, 16, and 19. By examining the pattern between the numbers:
  • The difference between the first two numbers is 1 (14 - 13).
  • The difference between the second and third is 2 (16 - 14).
  • The difference between the third and fourth is 3 (19 - 16).
This shows a progression where each difference increases by 1. Recognizing such a pattern helps us predict future numbers by continuing the established sequence of increasing differences.
This step of pattern recognition involves observing number sequences and identifying consistent ways they change.
Difference Calculation
Calculating the difference between numbers is a crucial step in understanding how a sequence progresses. We calculate the differences by subtracting each number from the next:
  • 14 - 13 = 1
  • 16 - 14 = 2
  • 19 - 16 = 3
These calculated differences indicate how the sequence evolves. They form their own mini-sequence: 1, 2, 3.
Noticing that this sequence of differences also progresses linearly means the next differences will logically continue this pattern. The next steps involve adding 4, then 5, and then 6 to the last known number in the list of terms.
Progression Analysis
Progression analysis refers to the process of extending the pattern to find additional terms in the sequence. By recognizing the pattern and calculating differences, we establish a clear roadmap for continuing the sequence.
Using the sequence established:
  • The next term, using a difference of 4, is 19 + 4 = 23.
  • Continuing, the difference of 5 gives us 23 + 5 = 28.
  • Finally, applying a difference of 6 results in 28 + 6 = 34.
Each step involves extrapolating from the previous pattern of differences, confirming our expected outcomes at every stage.
By understanding this progression approach, students can confidently find new terms in any similar arithmetic sequence.

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Most popular questions from this chapter

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the line through points at \((-5,-2)\) and \((1,2)\)

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